71 research outputs found
Between Treewidth and Clique-width
Many hard graph problems can be solved efficiently when restricted to graphs
of bounded treewidth, and more generally to graphs of bounded clique-width. But
there is a price to be paid for this generality, exemplified by the four
problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that
are all FPT parameterized by treewidth but none of which can be FPT
parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7,
8]. We therefore seek a structural graph parameter that shares some of the
generality of clique-width without paying this price. Based on splits, branch
decompositions and the work of Vatshelle [18] on Maximum Matching-width, we
consider the graph parameter sm-width which lies between treewidth and
clique-width. Some graph classes of unbounded treewidth, like
distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph
Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized
by sm-width
Group Activity Selection with Few Agent Types
The Group Activity Selection Problem (GASP) models situations where a group of agents needs to be distributed to a set of activities while taking into account preferences of the agents w.r.t. individual activities and activity sizes. The problem, along with its well-known variants sGASP and gGASP, has previously been studied in the parameterized complexity setting with various parameterizations, such as number of agents, number of activities and solution size. However, the complexity of the problem parameterized by the number of types of agents, a natural parameter proposed already in the first paper that introduced GASP, has so far remained unexplored. In this paper we establish the complexity map for GASP, sGASP and gGASP when the number of types of agents is the parameter. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps which allow us to focus on solutions which are "acyclic". These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta, Roy, Saurabh and Zehavi (2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well
A Single-Exponential Fixed-Parameter Algorithm for Distance-Hereditary Vertex Deletion
Vertex deletion problems ask whether it is possible to delete at most
vertices from a graph so that the resulting graph belongs to a specified graph
class. Over the past years, the parameterized complexity of vertex deletion to
a plethora of graph classes has been systematically researched. Here we present
the first single-exponential fixed-parameter tractable algorithm for vertex
deletion to distance-hereditary graphs, a well-studied graph class which is
particularly important in the context of vertex deletion due to its connection
to the graph parameter rank-width. We complement our result with matching
asymptotic lower bounds based on the exponential time hypothesis. As an
application of our algorithm, we show that a vertex deletion set to
distance-hereditary graphs can be used as a parameter which allows
single-exponential fixed-parameter tractable algorithms for classical NP-hard
problems.Comment: 43 pages, 9 figures (revised journal version; an extended abstract
appeared in the proceedings of MFCS 2016
Model Counting for Formulas of Bounded Clique-Width
We show that #SAT is polynomial-time tractable for classes of CNF formulas
whose incidence graphs have bounded symmetric clique-width (or bounded
clique-width, or bounded rank-width). This result strictly generalizes
polynomial-time tractability results for classes of formulas with signed
incidence graphs of bounded clique-width and classes of formulas with incidence
graphs of bounded modular treewidth, which were the most general results of
this kind known so far.Comment: Extended version of a paper published at ISAAC 201
Expanding the expressive power of Monadic Second-Order logic on restricted graph classes
We combine integer linear programming and recent advances in Monadic
Second-Order model checking to obtain two new algorithmic meta-theorems for
graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of
the well-known Monadic Second-Order logic by the addition of cardinality
constraints, can be solved in FPT time parameterized by vertex cover. The
second meta-theorem shows that the MSO partitioning problems introduced by Rao
can also be solved in FPT time with the same parameter. The significance of our
contribution stems from the fact that these formalisms can describe problems
which are W[1]-hard and even NP-hard on graphs of bounded tree-width.
Additionally, our algorithms have only an elementary dependence on the
parameter and formula. We also show that both results are easily extended from
vertex cover to neighborhood diversity.Comment: Accepted for IWOCA 201
The Power of Cut-Based Parameters for Computing Edge-Disjoint Paths
This paper revisits the classical edge-disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter tree-cut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded tree-cut width. Our second result shows that EDP parameterized by tree-cut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph
The complexity landscape of decompositional parameters for ILP
Integer Linear Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems in artificial intelligence are solved in practice via a translation to ILP. Despite its huge range of applications, only few tractable fragments of ILP are known, probably the most prominent of which is based on the notion of total unimodularity. Using entirely different techniques, we identify new tractable fragments of ILP by studying structural parameterizations of the constraint matrix within the framework of parameterized complexity.
In particular, we show that ILP is fixed-parameter tractable when parameterized by the treedepth of the constraint matrix and the maximum absolute value of any coefficient occurring in the ILP instance. Together with matching hardness results for the more general parameter treewidth, we give an overview of the complexity of ILP w.r.t. decompositional parameters defined on the constraint matrix
On the Hardness of SAT with Community Structure
Recent attempts to explain the effectiveness of Boolean satisfiability (SAT)
solvers based on conflict-driven clause learning (CDCL) on large industrial
benchmarks have focused on the concept of community structure. Specifically,
industrial benchmarks have been empirically found to have good community
structure, and experiments seem to show a correlation between such structure
and the efficiency of CDCL. However, in this paper we establish hardness
results suggesting that community structure is not sufficient to explain the
success of CDCL in practice. First, we formally characterize a property shared
by a wide class of metrics capturing community structure, including
"modularity". Next, we show that the SAT instances with good community
structure according to any metric with this property are still NP-hard.
Finally, we study a class of random instances generated from the
"pseudo-industrial" community attachment model of Gir\'aldez-Cru and Levy. We
prove that, with high probability, instances from this model that have
relatively few communities but are still highly modular require exponentially
long resolution proofs and so are hard for CDCL. We also present experimental
evidence that our result continues to hold for instances with many more
communities. This indicates that actual industrial instances easily solved by
CDCL may have some other relevant structure not captured by the community
attachment model.Comment: 23 pages. Full version of a SAT 2016 pape
On the complexity landscape of connected f-factor problems
Let G be an undirected simple graph having n vertices and let f:V(G)→{0,…,n−1} be a function. An f-factor of G is a spanning subgraph H such that dH(v)=f(v) for every vertex v∈V(G). The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte (Can J Math 6(1954):347–352, 1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connectedf-factor is easily seen to generalize Hamiltonian Cycle and hence is NP-complete. In fact, the Connected f-Factor problem remains NP-complete even when we restrict f(v) to be at least nϵ for each vertex v and constant 0≤ϵ1, the problem is NP-intermediate
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
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